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Comparative analysis of lump, breather, and interaction solutions using a bidirectional data mapping approach.

Syeda Sarwat Kazmi1,2, Muhammad Bilal Riaz3,4

  • 1IT4Innovations, VSB - Technical University of Ostrava, Ostrava, Czech Republic. syeda.sarwat.kazmi@vsb.cz.

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Summary

This study uses the Hirota bilinear method to find nonlinear wave solutions for the Boussinesq equation, crucial for coastal engineering. It classifies soliton, breather, and lump waves, analyzing their elastic interactions for predicting rogue waves.

Keywords:
[Formula: see text]-soliton solutionsBreather wavesHirota bilinear methodHybrid solutionsLong wave limit methodLump wavesOverlapping of solutions

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Area of Science:

  • Nonlinear Dynamics
  • Fluid Mechanics
  • Coastal and Ocean Engineering

Background:

  • The [Formula: see text]-dimensional Boussinesq equation models long wave propagation in shallow water, essential for coastal and ocean engineering.
  • Understanding nonlinear wave structures is key to predicting energy localization, wave stability, and extreme events like rogue waves.

Purpose of the Study:

  • To derive explicit [Formula: see text]-soliton, breather, and lump wave solutions for the [Formula: see text]-dimensional Boussinesq equation.
  • To analyze the interaction dynamics of these nonlinear waves and develop hybrid solutions.
  • To provide a unified framework for identifying conditions leading to similar wave dynamics using a bidirectional scatter plot technique.

Main Methods:

  • Hirota bilinear method for deriving explicit [Formula: see text]-soliton solutions (bright and dark types).
  • Complex conjugate approach for constructing breather solutions.
  • Long-wave limit method for obtaining first- and second-order lump waves.
  • Analysis of interaction dynamics, including soliton-soliton and soliton-lump collisions.
  • Application of a bidirectional scatter plot technique for comparative analysis.

Main Results:

  • Explicit classification of [Formula: see text]-soliton solutions into bright and dark types.
  • Construction of breather solutions in different planes and lump waves (rationally localized structures).
  • Development of four hybrid solutions combining solitons, lumps, and breathers.
  • Demonstration that wave interactions are elastic, with structures retaining identity post-collision.
  • Identification of conditions for stable energy concentrations (lump solutions) and oscillatory instabilities (soliton-breather interactions).

Conclusions:

  • The study provides a systematic taxonomy of nonlinear wave structures for the Boussinesq equation.
  • Developed solutions and interaction analyses offer practical insights for predicting wave behavior and extreme events in oceanic conditions.
  • The bidirectional scatter plot technique serves as a novel diagnostic tool for engineers evaluating wave interactions.